Helpl with a few questions (1 Viewer)

[mreditor16]

So, I've started MATH at uni, and they have given us some Qs which 'revises' 3U and 4U content. I am stuck on a few, so any help on these would be much appreciated.

1. At the time of printing, the largest known prime number (found in August 2008) is (2^43112609) - 1

How many digits are in its decimal expansion (hint - think log base 10)

2. By taking cases or otherwise, prove that 1 + x +x^2 + x^3 + x^4 is always positive.

As I go along, I might have a few more, which I will update this post with!

Thanks a lot!

 

[faisalabdul16]

Provide the cases...

The original question tells you what cases to consider

 

[InteGrand]

So, I've started MATH at uni, and they have given us some Qs which 'revises' 3U and 4U content. I am stuck on a few, so any help on these would be much appreciated.

1. At the time of printing, the largest known prime number (found in August 2008) is (2^43112609) - 1

How many digits are in its decimal expansion (hint - think log base 10)

2. By taking cases or otherwise, prove that 1 + x +x^2 + x^3 + x^4 is always positive.

As I go along, I might have a few more, which I will update this post with!

Thanks a lot!

1. By inspection, the number of digits in the decimal expansion of an integer N > 0 is . (e.g. number of digits in 10^3.4421 is 3+1=4 (ignoring the fractional part of the number)).

Therefore, the number of digits in the decimal expansion of is:



, since the number of digits in the decimal expansion of equals that of , since for positive integers n, k

(log laws)





.

 

[mreditor16]

Provide the cases...

The original question tells you what cases to consider

Oh haha yep, forgot to include them.

The cases are x >= 0 , -1 < x < 0 and x<= -1

 

[mreditor16]

1. By inspection, the number of digits in the decimal expansion of an integer N > 0 is . (e.g. number of digits in 10^3.4421 is 3+1=4 (ignoring the fractional part of the number)).

Are you expected to pluck that out of nowhere? Like how do you realise that to be your first step?

 

[InteGrand]

So, I've started MATH at uni, and they have given us some Qs which 'revises' 3U and 4U content. I am stuck on a few, so any help on these would be much appreciated.

1. At the time of printing, the largest known prime number (found in August 2008) is (2^43112609) - 1

How many digits are in its decimal expansion (hint - think log base 10)

2. By taking cases or otherwise, prove that 1 + x +x^2 + x^3 + x^4 is always positive.

As I go along, I might have a few more, which I will update this post with!

Thanks a lot!

2. For any , the given polynomial is obviously positive. By the sum of a geometric series, the given polynomial equals . Clearly, for any x < 0, this has a negative numerator, and a negative denominator, so the overall fraction (and hence the polynomial) is positive.

Hence the given polynomial is positive for all real x.

 

[InteGrand]

Are you expected to pluck that out of nowhere? Like how do you realise that to be your first step?

Well, when a number equals "10^something", it's like 1, with "something" 0's after it, which is "something +1" digits in base 10. This "something" can be written as , where . So

.

Clearly, it's the that controls the number of digits, which will then be as mentioned earlier. (The controls what those digits are.)

And this "something" is just , since

 

[mreditor16]

Got one more question:



I keep getting 0.863163549....

but the answers say 0.5

:/

 

[InteGrand]

Got one more question:



I keep getting 0.863163549....

but the answers say 0.5

:/

I guess the answers are wrong, it should be what you got .

Maybe when the answers were written, they forgot they asked for and thought they'd asked for , which would be 0.5.

Edit: wait, I might be wrong

Edit 2: Yeah, it's 0.5.

 

[enigma_1]

Provide the cases...

The original question tells you what cases to consider

welcome back

 

[mreditor16]

I guess the answers are wrong, it should be what you got .

Maybe when the answers were written, they forgot they asked for and thought they'd asked for , which would be 0.5.

Edit: wait, I might be wrong

Edit 2: Yeah, it's 0.5.

Wait how, and I got 1 / (2 - sin(1)) keeping in mind radians mode.

 

[InteGrand]

(0,1) lies on the graph of f(x), so the corresponding point on the graph of g(x) is (1,0), and at this point, the slope is the reciprocal of that on (0,1) of f(x), which is .

 

[InteGrand]

Wait how, and I got 1 / (2 - sin(1)) keeping in mind radians mode.

Sorry, I meant sin (typo)

 

[InteGrand]

So if and are inverse functions, and , then , provided .

 

[mreditor16]

(0,1) lies on the graph of f(x), so the corresponding point on the graph of g(x) is (1,0), and at this point, the slope is the reciprocal of that on (0,1) of f(x), which is .

ohhhh, I got an expression for derivative of g(x) in previous post, but subbed in x=1 instead of y=0 oops thanks

 

[mreditor16]

Can anyone help me with part c)?

 

[buriza]

Can anyone help me with part c)?

Reposting from the other thread.

 

[mreditor16]

Reposting from the other thread.

Thanks Buriza!